-
A Comparison of Imaging Methods using GPR for Landmine
Detection
and A Preliminary Investigation into the SEM for
Identification
of Buried Objects
A Thesis Presented
To the Faculty of Graduate Studies
By
Colin Gerald Gilmore
In Partial Fulfillment
of the Requirements for the Degree of
Masters of Science in Electrical Engineering
University of Manitoba
Winnipeg, Manitoba, Canada
December 2004
© Colin Gerald Gilmore
-
AbstractPart I
Various image reconstruction algorithms used for subsurface
targets are reviewed. It
is shown how some approximate wavefield inversion techniques:
Stripmap Synthetic
Aperture Radar (SAR), Kirchhoff Migration (KM) and
Frequency-Wavenumber (FK)
migration are developed from various models for wavefield
scattering. The similarities of
these techniques are delineated both from a theoretical and
practical perspective and it is
shown that Stripmap SAR is, computationally, almost identical to
FK migration. A plane
wave interpretation of both Stripmap SAR and FK migration is
used to show why they are
so similar. The electromagnetic assumptions made in the image
reconstruction algorithms
are highlighted. In addition, it is shown that, theoretically,
FK and KM are identical.
Image reconstruction results for KM, Stripmap SAR and FK are
shown for both synthetic
and experimental Ground Penetrating Radar (GPR) data.
Subjectively the reconstructed
images show little difference, but computationally, Stripmap SAR
(and therefore, FK
migration) are much more efficient.
Part II
A preliminary investigation into the use of the Singularity
Expansion Method
(SEM) for use in identifying landmines is completed using a
Finite-Difference Time-
Domain code to simulate a simplified GPR system. The Total Least
Squares Matrix Pencil
Method (TLS-MPM) is used to determine the complex poles from an
arbitrary late-time
signal. Both dielectric and metallic targets buried in lossless
and lossy half-spaces are con-
sidered. Complex poles (resonances) of targets change
significantly when the objects are
buried in an external medium, and perturbation formulae for
Perfect Electric Conductor
(PEC) and dielectric targets are highlighted and used. These
perturbation formulae are
developed for homogenous surrounding media, and their
utilization for the half-space
i
-
(layered medium) GPR problem causes inaccuracies in their
predictions. The results show
that the decay rate (real part) of the complex poles is not
suitable for identification in this
problem, but that with further research, the resonant frequency
(imaginary part) of the
complex poles shows promise as an identification feature.
Keywords: GPR, Seismic Migration, Stripmap SAR, Landmines,
Target Identifica-
tion, SEM
ii
-
iii
ContributionsPart I
The main contribution of Part I is to show why Stripmap SAR and
Frequency Wave-
number migration are computationally almost identical. Key to
the comparison of both
techniques is a plane wave interpretation of both. To the
authors knowledge, an electro-
magnetics based plane wave interpretation for Stripmap SAR has
not been completed pre-
viously. In addition, it is not well understood in the radar
community that Kirchhoff
Migration and FK migration are equivalent from a theoretical
perspective, and this is also
shown.
Part II
The contribution of Part II lies in the application of the
perturbation formulae devel-
oped by Baum [30] and Hanson [32] for PEC and dielectric
targets, respectively. The PEC
formula has been used previously for UXO targets, but not for
landmine targets. No pub-
lished attempt of the use of the dielectric perturbation formula
has been seen by the author.
-
iv
AcknowledgementsThe author would like to thank his research
advisor Dr. Joe LoVetri for his encour-
agement, support, guidance, and extra effort in the preparation
of this manuscript. This
project would not have been possible without his continuous
support.
Thanks to Ian Jeffrey and Hong Su for help in the implementation
of the Stripmap
SAR algorithm. Many thanks to Eduardo Corral and Daniel Flores
for their aid in develop-
ing the experimental GPR system, and thanks to Michael Phelan
for developing the sand-
box testing environment.
-
v
AcronymsBOR -Body of RevolutionCNR -Complex Natural ResonanceDFT
-Discrete Fourier TransformDIFT Discrete Inverse Fourier
TransformEM - ElectromagneticsFDTD - Finite Difference Time
DomainFFT - Fast Fourier Transform
FK - Frequency-Wavenumber ( )FT - Fourier TransformGPIB -
General Purpose Instrumentation BusGPR - Ground Penetrating RadarHS
- Hyperbolic SummationIFT - Inverse Fourier TransformKM - Kirchhoff
MigrationMPM - Matrix Pencil MethodPEC - Perfect Electric
ConductorSAR - Synthetic Aperture RadarSEM - Singularity Expansion
MethodSVD - Singular Value DecompositionTLS - Total Least SquaresUN
- United NationsVNA - Vector Network Analyzer
ω k–
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Table of ContentsAbstract
..............................................................................................................................
i
Contributions
...................................................................................................................
iii
Acknowledgements..........................................................................................................
iv
Acronyms..........................................................................................................................
v
Table of
Contents.............................................................................................................
vi
List of Figures
...................................................................................................................
x
List of Tables
.................................................................................................................
xiii
Chapter 1Introduction and Motivation
............................................................................................
1
1.1 Problem Discussion
................................................................................
11.2 The Inverse Problem in Electromagnetics
.............................................. 21.3 Basic GPR
Principles
..............................................................................
41.4 Thesis Overview
.....................................................................................
5
Chapter 2Electromagnetic and Fourier Transform Concepts
.......................................................... 7
2.1 Basic EM Concepts
.................................................................................
72.1.1 The Wave Equation
....................................................................
82.1.2 Seismics and Electromagnetics
.................................................. 9
2.2 The General Plane Wave
......................................................................
102.3 Green’s Functions and the Kirchhoff Integral Equation
....................... 11
2.3.1 Green’s
Functions.....................................................................
112.3.2 The Kirchhoff Integral Formula
............................................... 12
2.4 The Continous Fourier Transform
........................................................ 132.5 The
Discrete Fourier Transform
........................................................... 13
2.5.1 Calculating DFT Parameters
.................................................... 142.5.2
Implementation of DFT and IDFT utilizing a SFCW Radar.... 16
Chapter 3Stripmap Synthetic Aperture Radar
...............................................................................
17
vi
-
3.1 Basic GPR Imaging Terminology
......................................................... 183.1.1
A-scans
.....................................................................................
183.1.2 B-scans, Range and
Cross-Range............................................. 193.1.3
C-scans
.....................................................................................
20
3.2 One Dimensional Range Profiling
........................................................ 213.2.1
Single Point Target from Two Locations
................................. 223.2.2 Target Hyperbolas and
B-scans................................................ 24
3.3 The Stripmap SAR Algorithm
..............................................................
253.3.1 The Interpolation Problem in Stripmap SAR
........................... 273.3.2 Graphical Representation of The
Stripmap SAR Algorithm.... 28
3.4 Electromagnetic Assumptions in Stripmap SAR
.................................. 283.4.1 The Vector Wavenumber in
Stripmap SAR............................. 293.4.2 The Exploding
Source Model...................................................
303.4.3 A Plane Wave Interpretation of Stripmap SAR Algorithm.....
30
Chapter 4Seismic Migration
..........................................................................................................
33
4.1 The Exploding Source Model
...............................................................
344.2 Hyperbolic Summation
.........................................................................
364.3 Kirchhoff Migration
..............................................................................
374.4 Frequency-Wavenumber Migration
...................................................... 41
Chapter 5Comparison of Imaging Techniques
..............................................................................
45
5.1 Theoretical Comparisons of Different Focusing Algorithms
............... 455.1.1 The Equivalence of Kirchhoff Migration and
Frequency-Wavenumber Migration
.................................................. 455.1.2
Similarities and Differences of F-K Migration and Stripmap
SAR....................................................................................
47
5.2 Practical Implementation of Imaging Algorithms
................................ 485.2.1
B-Scans....................................................................................
495.2.2 Stripmap SAR/F-K Migration
.................................................. 505.2.3
Kirchhoff Migration
.................................................................
52
5.3 Experimental Setup
...............................................................................
545.3.1 Synthetic Data
Generation........................................................
545.3.2 Experimental Data
....................................................................
55
5.4 Results
...................................................................................................
565.4.1 Synthetic Data
Results..............................................................
575.4.2 Experimental
Results................................................................
61
vii
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5.5 Conclusions and Future Work of Part I
................................................ 665.5.1 Future
Work..............................................................................
67
Chapter 6Introduction to Part II
....................................................................................................
69
6.1 The Singularity Expansion Method for Detection of Landmines
......... 706.1.1 Finding Poles from an Arbitrary Signal
................................... 71
6.2 The SEM Method for Buried Targets
................................................... 726.2.1 Buried
PEC Targets
..................................................................
726.2.2 Buried Dielectric Targets
......................................................... 726.2.3
Discussion of Buried Targets in
General.................................. 73
6.3 Implementation of the SEM Method for GPR
...................................... 756.3.1 FDTD Simulation of
GPR ........................................................ 76
Chapter 7The Singularity Expansion Method
...............................................................................
77
7.1 Introduction to the SEM
........................................................................
787.1.1 SEM From a Signal Processing Perspective
............................ 797.1.2 Determining the Onset of
Late-Time ....................................... 81
7.2 Perturbation Formula For Buried PEC Targets
.................................... 827.3 Perturbation For
Dielectric Targets
...................................................... 837.4 The
Matrix Pencil Method
....................................................................
85
Chapter 8Results and Conclusions from Part 2
.............................................................................
88
8.1 Total Field Formulation FDTD Generation of Data
............................. 898.1.1 Generation of FDTD Data
........................................................ 92
8.2 Experiments 1 and 2: Twenty Centimeter PEC Wire
........................... 938.2.1 Free Space Poles of 20 cm
Wire............................................... 938.2.2 Twenty
cm Wire Buried in Lossless Ground ...........................
978.2.3 Twenty cm Wire Buried in Lossy Ground
............................... 99
8.3 PEC Landmine
....................................................................................
1018.3.1 PEC Landmine Buried in Lossless Medium
.......................... 1028.3.2 PEC Landmine in Lossy Medium
.......................................... 105
8.4 Dielectric Landmine-Like Targets
...................................................... 1068.4.1
Calculation of Internal Resonances and Perturbations ...........
1078.4.2 Dielectric Landmine in Lossless Ground
............................... 1078.4.3 Dielectric Landmine Number
2 .............................................. 110
8.5 Conclusions and Future Work
............................................................
112
viii
-
8.5.1 Future
Work............................................................................
113
Chapter 9Conclusions
..................................................................................................................
115
Appendix AInterpolation in Stripmap SAR and Frequency-Wavenumber
Migration .................... 116
Appendix BSource Code Examples
................................................................................................
122
b.1 Source Code for Part I
.........................................................................
122b.1.1 Matlab Source Code for Stripmap SAR, F-K Migrationand
B-scans......................................................................................
122b.1.2 Kirchhoff Migration Source
Code.......................................... 130
b.2 Source Code for Part II
.......................................................................
133b.2.1 FDTD Source Code for PEC 20cm Wire
............................... 133b.2.2 Matlab Source Code for the
Matrix Pencil Method ............... 134
References.....................................................................................................................
138
ix
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List of FiguresFigure 1.1 : The Forward and Inverse Problem in
Electromagnetics .............................. 2
Figure 1.2 : The Basic GPR System
................................................................................
4
Figure 2.1 :The Time Domain Discrete Fourier Transform
.......................................... 15
Figure 2.2 : Zero Padding of SFCW Radar Data
........................................................... 16
Figure 3.1 : Example of A-Scan
....................................................................................
19
Figure 3.2 : Example of
B-Scan.....................................................................................
20
Figure 3.3 : A 1-D Plane Wave Illuminating a Perfectly
Conducting Half Space ........ 21
Figure 3.4 : Range Imaging of Single Point Target from Two
Locations ..................... 22
Figure 3.5 : Range Profiles of a Single Target from Two
Positions.............................. 23
Figure 3.6 : Graphical Representation of How Target Hyperbola
Occurs .................... 24
Figure 3.7 :Stripmap SAR Setup
...................................................................................
25
Figure 3.8 : Graphical Representation of the 2-D Stripmap SAR
Algorithm................ 28
Figure 3.9 : Plane Wave Interpretation of Stripmap SAR
............................................. 31
Figure 4.1 : The Exploding Source Model
....................................................................
35
Figure 4.2 : Kirchhoff Migration Coordinate
System.................................................... 37
Figure 4.3 : Graphical Representation of 2-D KM
........................................................ 41
Figure 4.4 : Graphical Representation of the 2-D Frequency-
Wavenumber Migration 44
Figure 5.1 : Implementation of a B-scan
.......................................................................
49
Figure 5.2 : Practical Implementation of Stripmap SAR and F-K
Migration Flow Chart 51
Figure 5.3 :Practical Implementation of KM Flow
Chart.............................................. 53
Figure 5.4 : Experimental Setup
....................................................................................
55
Figure 5.5 : Photograph of Experimental Setup
............................................................ 56
Figure 5.6 : Unfocused Synthetic Data for Two Point Targets
..................................... 57
Figure 5.7 : Stripmap SAR Focused Synthetic Data for Two Point
Targets ................. 58
Figure 5.8 : F-K Migration Focused Synthetic Data for Two Point
Targets ................. 59
x
-
Figure 5.9 : KM Focused Synthetic Data for Two Point Targets
.................................. 59
Figure 5.10 : Experimental Data Layout
.......................................................................
62
Figure 5.11 : Unfocused Data for 0.8-5GHz SFCW Radar
........................................... 63
Figure 5.12 : Stripmap SAR Focused Image for 0.8-5GHz SFCW Radar
.................... 64
Figure 5.13 : F-K Migration Focused Image for 0.8-5GHz SFCW
Radar .................... 64
Figure 5.14 : KM Focused Image for 0.8-5GHz SFCW Radar
..................................... 65
Figure 5.15 : Difference Between Stripmap SAR and F-K Focused
Images for 0.8-5GHz SFCW Radar
..................................................................................................................
66
Figure 6.1 : Poles for Wire in Half-Space
.....................................................................
74
Figure 7.1 : Scattering of an Incident Wave by a Target
............................................... 78
Figure 7.2 : Early and Late Time Representation of a
Signal........................................ 81
Figure 8.1 : FDTD GPR
System....................................................................................
89
Figure 8.2 : Time Domain Wave Form of Derivative of Gaussian
Source ................... 91
Figure 8.3 : Magnitude of Frequency Domain of FDTD
Source................................... 92
Figure 8.4 : Ex from 20 cm x 1 cm x 1 cm PEC Wire in Free Space
............................ 94
Figure 8.5 : Magnitude of Singular Values for 40 cm PEC
Wire.................................. 95
Figure 8.6 : Poles for Free Space 20cm PEC
Wire........................................................ 96
Figure 8.7 : Ex from 20 cm x 1 cm x 1 cm PEC Wire Buried in
Lossless Media ......... 97
Figure 8.8 : Poles for 20 cm Wire Buried in a Lossless
Medium.................................. 98
Figure 8.9 : 20 cm Wire Buried in Lossy
Medium......................................................
100
Figure 8.10 : Poles for 20 cm Wire Buried in Lossy
Medium..................................... 101
Figure 8.11 : Free Space Response from PEC Landmine at T1
.................................. 102
Figure 8.12 : PEC Landmine Buried in Lossless Medium
.......................................... 103
Figure 8.13 : Poles for Buried Landmine in Lossy Medium
....................................... 104
Figure 8.14 : Poles for PEC Landmine in Lossy
Medium........................................... 105
Figure 8.15 : Time Domain Response of Dielectric Landmine In
Lossless Medium . 108
Figure 8.16 : Poles for Dielectric Landmine in Lossless Medium
.............................. 110
Figure 8.17 : Time Domain Response of Dielectric Landmine #2 in
Lossy Ground .. 111
Figure 8.18 : Poles for Dielectric Landmine #2 in Lossless
Medium ......................... 112
Figure a.1 : Discrete Data in Stripmap
SAR................................................................
117
Figure a.2 : Discrete Data in the Regular Frequency and Spatial
Frequency Domain 118
xi
-
Figure a.3 : Discrete Stripmap SAR data in Spatial Frequency
Domains ................... 119
xii
-
xiii
List of TablesTable 5.1: Comparison of Focusing Algorithms for
Synthetic Data ............................. 61
Table 8.1: List of FDTD Experiments
Conducted.........................................................
93
Table 8.2: PEC Wire Free-Space Pole Locations
.......................................................... 96
Table 8.3: PEC Wire in Lossless Medium Pole
Locations............................................ 98
Table 8.4: PEC Wire in Lossy Medium Pole Location
............................................... 100
Table 8.5: PEC Landmine in Lossless Medium Pole
Locations.................................. 103
Table 8.6: PEC Landmine in Lossy Medium Pole
Locations...................................... 105
Table 8.7: Dielectric Landmine #1 in Lossless Medium Pole
Locations .................... 108
Table 8.8: Dielectric Landmine #2 in Lossy Medium Pole Locations
........................ 111
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Chapter 1Introduction and Motivation
In this chapter, the landmine problem is introduced, as are
basic Ground Penetrating
Radar (GPR) and inverse imaging concepts. The basic concept of
synthetic aperture radar/
migration is introduced, as is the singularity expansion
method.
1.1 Problem Discussion
The United Nations (UN) estimates that there are around 110
million buried land-
mines throughout the world. These cause the deaths of
approximately 15,000 to 20,000
people each year. Most (80%) of these deaths are civilians. As
they provide a cheap and
effective tool of war, their use is extremely widespread:
casualties were reported in 65
countries in 2002. [1]
The ‘Ottawa Treaty’, which banned the use, distribution and
manufacture of land-
mines was signed by 122 countries in 1997, and has been ratified
by over 40. However,
the clearing of past landmines is still a large humanitarian
issue, and the number of land-
mines cleared each year is outnumbered by the number of new
landmines laid by a factor
of 20 to 1 [1]. While the best way to deal with the global
landmine problem is clearly to
continue the efforts of the Ottawa Treaty and stop their use,
the clearance of previously
laid landmines remains a serious humanitarian issue.
In order to deem land as ‘usable’, 99.6% of landmines buried up
to 10 cm deep must
be removed. Current landmine systems, which include prodding,
metal detection, and
sniffer dogs are slow and dangerous. In the late 1990’s there
was a large amount of
research that went into landmine detection and removal programs
utilizing Ground Pene-
trating Radar (GPR). While many papers have been written (see,
for example, the semi-
1
-
annual GPR conference [2]), no fast, easy-to-use and safe system
has been developed. The
purpose of this thesis is to advance the currently developed
signal processing (imaging,
detection and identification) methods to better the detection
and removal of landmines.
1.2 The Inverse Problem in Electromagnetics
The two methods explored in this thesis are that of focusing
images generated by
GPR, termed migration of images, and that of applying the
Singularity Expansion Method
for target identification. Both of these methods are a subset of
the GPR problem, and the
GPR problem is a subset of what is known as the inverse
problem.
We can contrast the inverse problem with the forward problem.
The general physical
set-up of an electromagnetic scattering problem is shown in
Figure 1.1. In the forward
problem, the incident field, and the material parameters , are
known and
the scattered field is to be determined. In the inverse problem,
we have knowledge of the
incident field, and some (always limited) knowledge of the
scattered field,
, and we desire to infer some knowledge of the material
parameters, .
The uniqueness theorem (see Harrington [3]), tells us that, in
the forward problem, the
Hinc E, incHscat Escat,
ε µ σ, ,
Figure 1.1: The Forward and Inverse Problem in
Electromagnetics
Hinc Einc, ε µ σ, ,
Hinc E, incHscat Escat, ε µ σ, ,
2
-
scattered fields, , are unique. However, the inverse problem is
much more
complicated. The general inverse problem is an extremely
interesting and complex prob-
lem; as W.C. Chew [4] states: “The ability to infer information
on an object without direct
contact also expands man’s sensory horizon. No doubt, it is a
much sought after capabil-
ity”.
Two important questions for mathematicians concerning a problem
are: 1) is there a
solution and 2) if there is, is it unique? In inverse imaging,
we are also concerned with the
question 3) if there is a solution, is it stable? We define
stable to mean that an arbitrarily
small change in the input (measured scattered fields) will not
cause an arbitrarily large
change in the solution (material parameters). In GPR we have
certainty that a solution
exists because some physical media gave rise to the observed
scattered fields. The inverse
problem, with knowledge of the scattered fields at all points
over all frequencies
( ), outside the object of interest, is unique for two
dimensions [5]. This author
has not seen a proof of uniqueness for three dimensions. The
mathematical proof of the
uniqueness has been claimed to be one of the greatest
accomplishments of 20th century
mathematics.[7]
However, in all practical situations, we deal with non-unique
solutions because we
cannot practically receive all frequencies at all points in
space outside the object of inter-
est. In addition, the problem can be shown to be unstable: an
arbitrarily small change in
the input can cause an arbitrarily large perturbation of the
solution. Problems that are
unstable and non-unique are called ill-posed problems [5].
Due to the ill-posed nature of the inverse problem, solution
techniques often have to
make simplifications and assumptions in order to be feasible.
Typically, this means mak-
ing assumptions about the material parameters beforehand. This
can be an acceptable pro-
cess, for example in GPR we often know or can find many
parameters of the soil
surrounding possible targets.
The GPR problem is a special case of the generalized inverse
problem. Both parts of
this thesis, the Synthetic Aperture Radar (SAR)/Seismic
Migration section and the Singu-
larity Expansion Method (SEM) section can be viewed as subsets
of the inverse problem.
The SAR/Migration algorithms fall under what can be termed
inverse imaging. Inverse
Hscat Escat,
0 ω ∞<
-
imaging involves building some type of image of the target,
while the general inverse
problem may not (such as the utilization of the SEM).
1.3 Basic GPR Principles
The basic process of GPR is depicted in Figure 1.2. The GPR can
either be bistatic,
with both a transmit and receive antenna, or monostatic with a
single transmit and receive
antenna. The basic idea is to excite the ground and possible
targets with electromagnetic
energy, then attempt to infer properties of the ground and
targets from scattering informa-
tion obtained from the experiment.
One of the biggest problems in GPR is the large ground
reflection that occurs. With
a high dielectric contrast between the ground and air, this
allows for a very small percent-
age of the transmitted energy pass the interface, reflect off
the target, and pass the inter-
face again to reach the receiving antenna.
In the GPR problem considered here, the antennas operate in free
space, while the
ground material properties are given by and , and the target’s
properties are
εo µo,
ε1 µ1 σ1, ,ε2 µ2 σ2, ,
Tx Rx
z
y
x
Figure 1.2: The Basic GPR System
ε1 µ1, σ1
4
-
given by and . Typically, the antenna(s) are moved over the
target in a straight
line, which allows for some type of synthetic aperture
processing.
1.4 Thesis Overview
Part I of this thesis is concerned with the investigation of
image focusing techniques
for use with GPR for landmine detection. Following this
introduction, Chapter 2 intro-
duces some electromagnetic concepts as well as some Fourier
transform fundamentals.
Chapter 3 describes Stripmap Synthetic Aperture Radar (SAR),
which is followed by
Chapter 4, a discussion of Seismic Migration techniques. Chapter
5 provides and dis-
cusses experimental and synthetic data as well as discussing
theoretical comparisons of
Stripmap SAR and the Seismic migration techniques. The work of
part one is summarized
in Gilmore, et. al. [8].
Part two of this thesis outlines a preliminary investigation of
the use of the Singular-
ity Expansion Method (SEM) for the identification of landmines.
Chapter 6 introduces this
section and reviews the relevant literature, and Chapter 7 gives
an overview of the Singu-
larity Expansion Method, the perturbation formulae for buried
scatterers and also dis-
cusses the Matrix Pencil Method (MPM) as a method of determining
the complex
resonances of electromagnetic scatterers. Chapter 8 presents
Finite-Difference Time-
Domain (FDTD) generated results and tests the perturbation
formulae. Finally, Chapter 8
concludes the thesis.
ε2 µ2, σ2
5
-
6
Part I
A Comparison of Seismic Migration and Stripmap SAR Imaging
Methods
for Ground Penetrating Radar for Landmine Detection
-
Chapter 2Electromagnetic and Fourier Transform Concepts
In this chapter, important electromagnetic concepts are covered.
In addition, the
basic concepts of the Discrete Fourier Transform (DFT) in both
the time/frequency and
spatial/spatial frequency domains are highlighted.
2.1 Basic EM Concepts
All macroscopic electromagnetic (EM) phenomena are governed by
Maxwell’s
Equations:
, (2.1)
, (2.2)
, (2.3). (2.4)
Where is electric field intensity, is magnetic field intensity,
is electric flux den-
sity, is magnetic flux density, is current density, and is
electric charge density.
These equations imply the equation of continuity,
, (2.5)
and in linear media, the field densities are related to the flux
densities via the constitutive
relations:
, (2.6), (2.7). (2.8)
E r t,( )∇× t∂∂ B r t,( )–=
H r t,( )∇× J r t,( )t∂∂ D r t,( )+=
∇ B r t,( )⋅ 0=∇ D r t,( )⋅ ρ r t,( )=
E H D
B J ρ
∇ J r t,( )⋅t∂∂ ρ r t,( )–=
D εE=B µH=J σE=
7
-
Where is the dielectric constant or electric permittivity, is
the magnetic permeability,
and is the conductivity of the medium. Together, the three
parameters , and
determine how a medium reacts to electromagnetic fields. In this
document, all bold
parameters in the equations are vectors.
If we assume a time-harmonic dependency ( ), Maxwell’s equations
can be writ-
ten in their time-harmonic form as:
, (2.9), (2.10)
, (2.11), (2.12)
with the equation of continuity becoming
. (2.13)While not shown here, integral formulations of Maxwells
equations can also be
derived.
If we define the admittivity [3] of the medium as
(2.14)
and the impedivity of a medium as:
(2.15)then we can define the wavenumber as
. (2.16)
In lossless media, the wavenumber is purely imaginary, while in
lossy media has a wave-
number with a real component.
2.1.1 The Wave EquationConsider a source free (impressed
current), ), linear,
homogenous and isotropic region of space. We can then write the
frequency-domain Max-
well’s Equations (2.9) and (2.10) as:
(2.17). (2.18)
ε µ
σ ε µ σ
ejωt
E r ω,( )∇× jωB r ω,( )–=H r ω,( )∇× J r ω,( ) jωD r ω,( )+=
∇ B r ω,( )⋅ 0=∇ D r ω,( )⋅ ρ r ω,( )=
∇ J r ω,( )⋅ jωρ r ω,( )–=
ŷ σ jωε+=
ẑ jωµ=
k ẑŷ–=
J r t,( ) 0= ρ r t,( ) 0=
E∇× ẑH–=H∇× ŷE=
8
-
By taking the curl of the first equation, and making a
substitution of the second, we
can arrive at the equation
. (2.19)This is known as the complex vector wave equation. The
magnetic field also follows the
same equation:
. (2.20)If we utilize the definition of the Laplacian
operator
(2.21)and the fact that the divergence of both and are zero, we
can arrive at the vector
wave equations:
, (2.22)
. (2.23)A very important result which is used in this thesis is
that each rectangular compo-
nent of both and satisfy the complex scalar wave equation or
Helmholtz equation:
. (2.24)Here, can be replaced by any of the components or .
We can also write the wave equation in the time domain for a
source free region as:
. (2.25)
where , and are the material parameters in the medium.
2.1.2 Seismics and ElectromagneticsIf we assume that the earth
can be treated as an acoustic medium we may also apply
lossless ( ) wave equation in both the time and frequency-domain
for source free
regions (Zhdanov [5] and Oristaglio et. al. [7]). The difference
is that would represent
the wave pressure, a scalar quantity, and the term is replaced
with , the velocity
of sound in the medium.
E∇×∇× k2E– 0=
H∇×∇× k2H– 0=
∇2A ∇ ∇ A⋅( ) ∇ ∇ A××–=E H
∇2E k2E+ 0=
∇2H k2H+ 0=
E H
∇2ψ k2ψ+ 0=ψ Ex Ey Ez, , Hx Hy Hz, ,
ψ∇2 µεt2
2
∂
∂ ψ– µσ t∂∂ψ– 0=
µ ε σ
σ 0=
ψ
1 µε( )⁄ v
9
-
2.2 The General Plane Wave
In general, the most elementary type of plane wave can be
represented by the func-
tion (Stratton pg 363) [9]:
, (2.26)where represents a length away from the origin, and can
be written as:
, (2.27)
where is the unit normal in the direction of wave travel. We
can
define the vector wavenumber as
, (2.28)
so that the elementary wave function can be written as
. (2.29)We can also write in phasor form as
(2.30)
Where we have denoted the phasor term with an underscore, used
here for clarity, but
omitted later. When the non-phasor form of the wave function is
introduced into the time-
domain scalar wave equation (2.25), we can arrive at the
separation equation:
. (2.31)
Thus, if any three of and are determined, the fourth is given by
the separation
equation, (2.31).
In two-dimensional problems (with space variables and ),
equation (2.31)
becomes
. (2.32)
ψ e j– kh jωt+=h
h n R⋅ nxx nyy nzz+ += =
n nxax̂ ny+ ây nzâz+=
kn knxâx knyây knzâz+ + kxâx kyây kzâz+ += =
ψ e j– k R jωt+⋅=ψ
ψ e j– k R⋅=
kx2 ky
2 kz2+ + µεω2 j– µσω k2= =
kx ky kz, , ω
y z
ky2 kz
2+ k2=
10
-
2.3 Green’s Functions and the Kirchhoff Integral Equa-tion
2.3.1 Green’s FunctionsThe basic ideas of Green’s functions are
presented here utilizing only a one-dimen-
sional problem (in time) for simplicity. Green’s functions used
in electromagnetics require
four-dimensions. More in-depth discussion of Green’s functions
in electromagnetics is
given in Morse and Feshbach [13] and D.S. Jones [14]. The aim
here is to aid the reader in
understanding the use of Green’s functions in the Kirchhoff
integral equation. The deriva-
tion presented here is based on Adomain [12] and many specific
details (such as initial
conditions of the operator equation and boundary values) are
sacrificed to keep the deriva-
tion clear.
If we consider an operator equation:
(2.33)where is a differential operator, is called the forcing
term and is the unknown solu-
tion to the operator equation. In addition to being a
differential operator, we also require
it to be a linear operator with an inverse. Utilizing the
existence of the inverse we may
write:
. (2.34)Utilizing the fact that is linear, and is a differential
operator, we can write the
solution as an integral type:
. (2.35)
This equation tells us that if we can find the unknown function
, which we
call the Green’s function, we can then find .
To provide a method of finding the Green’s function, we apply
the operator to
(2.35)
. (2.36)
Utilizing (2.33) and the linearity of , we can write
Lu t( ) f t( )=L f u
L
u t( ) L 1– f t( )=L
u t( ) L 1– f t( ) G t τ,( )f τ( ) τd∫= =G τ t,( )
u t( )
L
Lu t( ) L G t τ,( )f τ( ) τd∫=L
11
-
. (2.37)
The only way for this equation to hold is if
. (2.38)Thus, the Green’s function is the solution of the
operator equation when the forcing
term is the dirac delta function . It is important to note that
the boundary condi-
tions (and therefore the physical media) will play a part in the
solution of the Green’s
function. We can view this process as finding the solution to
the operator equation from a
point source, then completing a weighted summing in (2.36), with
the weighting function
equal to the forcing function .
For clarity, we have considered functions that only depend upon
the variable . In
electromagnetics, however, we must consider quantities that vary
with both space and
time. We therefore write the Green’s function in the form:
(2.39)
which we can write as the solution to the operator equation at
location and time
from a point source located at and time ‘. The rest of the terms
in the above derivation
can be similarly modified.
2.3.2 The Kirchhoff Integral FormulaThe Kirchhoff integral
formula is an integral solution of the scalar wave equation
(2.24). A derivation of the solution can be found in D.S. Jones
[14]. The formula is pre-
sented here without proof, and the associated boundary
conditions for the surface are
discussed in section 4.3. The Kirchhoff integral solution to the
scalar wave equation for a
source-free medium is
. (2.40)
Where is the scalar field solution we are attempting to solve
for, is the surface
of the volume in which we are attempting to find a solution, is
the outward normal to
, and represents the measurements of this scalar field.
f t( ) LG t τ,( )f τ( ) τd∫=
LG t τ,( ) δ t τ–( )=
δ t τ–( )
f t( )
t
G r t r′ t′,,( )
L r t
r′ t′
S′
ψ r t,( ) ψ r′ t,( )n′∂∂ G r t r′ t′,,( ) G r t r′ t′,,( )
n′∂∂ ψ r′ t′,( )–
S′∫° S′d t′d
t′∫–=
ψ r t,( ) S′
n′
S′ ψ r′ t,( )
12
-
2.4 The Continous Fourier Transform
In this thesis, the continous forward Fourier Transform (FT) of
a signal is
defined as
, (2.41)
and the inverse fourier transform (IFT) is defined as
. (2.42)
where is the Fourier-domain variable. If the variable represents
time then ,
the usual frequency-domain Fourier variable. For a spatial
variable such as , we use
and . An equivalent transform can be defined with a positive in
the expo-
nential term for the forward transform, and a negative in the
exponential term for the
reverse transform. Both versions of the transform will be used
later in this thesis.
In general, we use upper-case letters for the Fourier domain
signals, and lower-case
letters for non-fourier domain signals.
The continous FT has been thoroughly studied elsewhere and the
reader is referred
to any standard textbook such as [10].
2.5 The Discrete Fourier Transform
The Discrete Fourier Transform (DFT) is an approximation of the
continous fourier
transform in a discrete form. If we imagine that we have a
discrete -domain signal of
total samples with a sampling period of . Then instead of
(2.43)
we then have
f u( )
F ku( )12π
---------- f u( )ej– kuu ud
∞–
∞
∫≡
f u( ) 12π
---------- F ku( )ejkuu kud
∞–
∞
∫=
ku u ku ω=
x
u x= ku kx=
u N
∆u
F ku( )12π
---------- f u( )ej– kuu ud
0
N∆u
∫≡
13
-
. (2.44)
Note that is still a continous variable. We now sample it at a
set of discrete points
where and multiply the transform by
. is sampled over a set of points from to . The effects of
this
constant scaling term can be cancelled by appropriately defining
the inverse transform.
We then define the DFT as
(2.45)
The Inverse DFT (IFDT) can be defined as
. (2.46)
For a thorough description of the DFT, see Proakis and Manolakis
[11].
2.5.1 Calculating DFT ParametersThe imaging algorithms of part I
of this thesis rely heavily on both forward and
inverse DFT or collected data. It is important to understand the
relations between both
Fourier domains and this next section highlights the relations
between them.
Consider a sampled signal in the domain, sampled times from
to
. Here, we assume that N is always an odd number. N being even
will cause a
slight modification of these formulae. The separation between
samples is . If we take
the DFT of this signal, we get an point signal in the frequency
domain, which goes from
to (if is even, this does not hold). In this discussion, we take
the select
the normalized angular frequency to be from to (this is in
contrast to the previous
F ku( )12π
---------- f n∆u( )ejkun∆u– ∆u
n 0=
N 1–
∑=
kuku
m 2πm N 1–( )( )⁄ ∆u )= m 0 1 … N 1–, , ,=
2π ∆u( )⁄ ku k– umax kumax
F kum( ) f n∆u( )e
j2πm nN----–
n 0=
N 1–
∑=
f n∆u( ) 1N---- F ku
m( )ej2πm nN
----
m 0=
N 1–
∑=
u N u 0=
u Umax=
∆u
N
ku– max kumax N
π– π
14
-
set of equations, where we selected a range of to ). The
separation between samples
in the frequency domain is . This is graphically represented in
Figure 2.1.
Starting with the fact that sampled points contain
intervals:
and . (2.47)
Using the fact that the sample frequency, is
(2.48)
The Nyquist sampling theorem states that
. (2.49)
Now that we have an equation for relating and , we can find
and
, using (2.47). Doing this:
and (2.50)
0 2π
∆ku
Figure 2.1:The Time Domain Discrete Fourier Transform
∆u
x n∆u( )
Umax
N total Points
Discrete FT
N total Points
∆kuu
kumaxkumax–
ku
X m∆ku( )
Discrete IFT
N N 1–
Umax N 1–( )∆u= 2kumax N 1–( )∆ku=
Fsample
Fsample1∆u-------=
kumax2πFsample
2------------------------ π
∆u-------= =
kumax ∆u ∆kuUmax
∆ku2kumaxN 1–( )
----------------- 2πN 1–( )∆u
------------------------- 2πUmax------------= = =
15
-
. (2.51)
Using this set of equations ((2.50) and (2.51)) and the
accompanying figure, we can
relate all the desired quantities to each other, for both the
discrete fourier transform as well
as the discrete inverse fourier transform. To apply a
time/frequency domain analysis we
simply replace with and with .
2.5.2 Implementation of DFT and IDFT utilizing a SFCW RadarWhen
implementing the DFT and DIFT in the practical case of Stepped
Frequency
Continous Wave (SFCW) radar, we are confronted with the problem
of finite bandwidth
data. That is, we only collect data from to . In order for
(2.50) and (2.51) to
apply, we must have the full bandwidth data. To accomplish this
we zero pad the data back
to , and then construct the negative frequencies from the
complex conjugate of this
zero-padded data. This step does add processing time to the
algorithm. The zero-padding
procedure is shown graphically in.Figure 2.2
Umax N 1–( )∆uπ N 1–( )
kumax--------------------- 2π
∆ku---------= = =
u t ku ω
ωstart ωstop
ω 0=
ωstart ωstopω
F ω( )
ωstart ωstopω
F ω( )
Filling
zeros
ωstart ωstopω
F ω( )
zeroszeros
ωstop– ωstart–
Add Complex Conjugate
Figure 2.2: Zero Padding of SFCW Radar Data
16
-
Chapter 3Stripmap Synthetic Aperture Radar
In this chapter, we begin with one-dimensional range imaging and
show why focus-
ing of these images is desirable. We then present the basic
Stripmap Synthetic Aperture
Radar (SAR) algorithm, and then give a description of the
electromagnetic assumptions
used in the Stripmap SAR algorithm.
The concept of Synthetic Aperture Radar (SAR) arose from the
radar community
when considering the problem of collecting a large amount of
data from an aeroplane. The
term synthetic aperture is based on the concept of simulating a
significantly larger antenna
aperture through signal processing, rather than constructing a
very large antenna. The
advantages of this can be illustrated as follows.
The cross-range, or lateral resolution ( ) of a radar can be
approximated by the
equation (Soumekh [6])
, (3.1)
where is the operating wavelength, is the diameter of the
antenna, and is the dis-
tance from the target to the antenna. If an antenna has a
diameter of meter, an
operating wavelength of meter, and a target at the range of
meters, then
the lateral resolution is 50 meters. If, however, we move our
radar along a line, creating an
imaginary or effective aperture larger than the physical
aperture (diameter) of the antenna,
we can improve the cross range resolution significantly. For
example, if we moved our
radar 200 meters, the effective diameter (aperture) would become
and the lat-
eral resolution would become 0.25 meters.
The Stripmap SAR algorithm can be contrasted with what we call
conventional
SAR, where the focusing is completed by convolving the received
radar data with the
Lr
LrRλD-------=
λ D R
D 1=
λ 1= R 50=
Deff 200=
17
-
inverse of the point target response. The so-called conventional
SAR can be shown equiv-
alent to the Hyperbolic Summation (HS) algorithm of geophysics
(for the HS algorithm,
see section 4.2) [15]. The HS method has long ago been replaced
by more sophisticated
methods, and Stripmap SAR is one of these more advanced methods.
As such, conven-
tional SAR will not be covered in this thesis.
The algorithm is presented here as it is seen in the literature
[6]; it seems that the
background of the authors for the main references in Stripmap
SAR do not lie in electro-
magnetics, rather in signal processing. The standard
presentation of the algorithm is
devoid of any references to electromagnetics, and the purpose of
the last section of this
chapter is to delineate a list of electromagnetic assumptions
left implicit in the Stripmap
SAR algorithm. The three most important assumptions are that we
ignore interactions
within and between targets (we model only point scatterers), we
assume a scalar wave
field (i.e. the vectorial character of EM fields is ignored) and
we assume that we know the
material parameters of the external medium. This discussion
facilitates the later compari-
son of this algorithm with seismic imaging techniques of chapter
4.
The Stripmap SAR algorithm is used extensively in modern radar
systems, for
example the E-3 AWACS (Airborne Warning and Control System)
aeroplanes, NASA
space shuttles and satellite borne radars all implement some
variation of this algorithm [6].
3.1 Basic GPR Imaging Terminology
3.1.1 A-scansWhen dealing with GPR images, we can define several
different types of ‘scans’.
The so-called A-scan, which is also known as a range profile, is
the result of a single pulse
(or sweep of frequencies) of a radar. For this type of example,
we can imagine an antenna
transmitting a pulse, then plotting the magnitude of the voltage
received at the received
antenna. Knowing the speed of light in the medium of interest,
we could plot the return on
18
-
a spatial axis. A graphical example of the A-scan, with two
targets, is shown in Figure 3.1.
The A-scan shown might be a return that we would expect from a
GPR with the larger
peak being the ground reflection and the smaller peak being a
possible target.
3.1.2 B-scans, Range and Cross-RangeA B-scan is simply a
succession of stacked A-scans. Here, we can imagine that we
take a series of A-scans over a single line (we move the
antennas along the axis). At a
set distance, , we perform an A-scan. The best way to display
this resulting image is in
Figure 3.1: Example of A-Scan
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Typical A-Scan
Range (Meters)
Nor
mal
ized
Mag
nitu
de o
f Ret
urn
Sig
nal
y
∆y
19
-
a 2-D plot, with areas of increasing intensity representing
scattering objects. An example
of B-scan is shown in Figure 3.2. This figure does not directly
relate to Figure 3.1, but is a
B-scan from a single point target.
The range is defined as the distance away from the antenna, the
cross-range is the
distance that the antenna has moved (in Figure 1.2, it would be
the distance moved in the
direction). This B-scan gives the response expected from a point
target at location
). The reasons behind the hyperbolic shape of this B-scan will
be
explored more thoroughly in 3.2.2.
3.1.3 C-scansC-scans are defined as a series of stacked B-scans.
Due to the fact that our experi-
mental data collection can, at this time, only be done on a
single plane, C-scans will not be
used in this thesis.
Figure 3.2: Example of B-Scan
Cross Range [meters]
Ran
ge [m
eter
s]
Simulated B-Scan
0 0.5 1 1.5 2 2.5 3 3.5 4
0
1
2
3
4
5
6
y
y z,( ) 2 1.5,( )=
20
-
3.2 One Dimensional Range Profiling
To understand the imaging algorithms in this thesis, it is
important to first under-
stand basic 1-D imaging. We show here how a single A-scan or
range profile can be con-
structed. Consider the problem shown in Figure 3.3. This is the
case of a 1-D plane wave
impinging on a point target located at a distance D from the
transmitter. We model the
expected return signal , as:
(3.2)where is the amplitude of the illuminating plane wave, is a
frequency independent
measure of the reflectivity of the reflecting plane, is the
distance from the transmitter to
the conducting sheet, and is the speed of light in the
surrounding medium.
We now take the IFT of the signal , and the resulting
time-domain signal is
. (3.3)
This equation is a dirac delta function with a scaling term and
we note that only phase
information is used.
To then map this data into the spatial domain, we simply scale
the time axis using
the equation
0z
ρ
z D=
Figure 3.3: A 1-D Plane Wave Illuminating a Perfectly Conducting
Half Space
S ω( )
S k( ) ρAej– 2ω
c----D
=A ρ
D
c
S ω( )
s t( ) ρA2π
----------δ t 2Dc----–⎝ ⎠
⎛ ⎞=
21
-
. (3.4)
Thus, equation (3.4) would become
(3.5)
which is equivalent to
. (3.6)
Which is a scaled dirac delta function at the point , which
represents the physical dis-
tance from the antenna. Using this transformation, we can switch
from the time to distance
axis and vice versa.
3.2.1 Single Point Target from Two LocationsWe now consider
performing a range profile of a single target from multiple
antenna
positions. The problem setup is shown in Figure 3.4. Here, we
imagine that we complete a
z c t2---=
s z( ) ρA2π
----------δ 2zc----- 2D
c-------–⎝ ⎠
⎛ ⎞ ρA2π
----------δ 2c--- z D–( )⎝ ⎠⎛ ⎞= =
s z( ) ρA2π
----------δ z D–( )=
D
ρ
y
z
y1
y2
R1
R2
Figure 3.4: Range Imaging of Single Point Target from Two
Locations
22
-
range profile with the antenna at position , then complete
another range profile with the
antenna at position .
The received signals from each range profile, and are modelled
as
and , (3.7)
which in the time domain become
and . (3.8)
If we then map the time domain into the spatial domain , we
arrive at:
and . (3.9)
These two signals are shown in Figure 3.5.
Thus, performing a standard range profile on a target results in
a Dirac delta function
located on the axis at , the distance between the antenna and
target.
y1y2
S1 k( ) S2 k( )
S1 k( ) ρAej2kR1= S2 k( ) ρAe
j2kR2=
s1 t( )ρA2π
----------δ t2R1
c---------–⎝ ⎠
⎛ ⎞= s2 t( )ρA2π
----------δ t2R2
c---------–⎝ ⎠
⎛ ⎞=
z
s1 z( )ρA2π
----------δ z R1–( )= s2 t( )ρA2π
----------δ z R2–( )=
z
z
s1 z( )
s2 z( )z R1=
ρA2π
----------
z R2=
ρA2π
----------
Figure 3.5: Range Profiles of a Single Target from Two
Positions
z R
23
-
3.2.2 Target Hyperbolas and B-scansWe now consider the case of a
single point target where we are moving our antenna
along a path defined by , as shown in Figure 3.6. The distance
between antenna and
target, as
. (3.10)
When we perform a range profile of this target at the distance ,
we get a Dirac
delta function at . We can then create an image by stacking all
these range profiles
together, resulting in a B-scan. Thus in our image,
. (3.11)
This describes a half-hyperbola in the plane:
. (3.12)
The hyperbola phenomena seen here gives rise to SAR focusing
algorithms, as point
targets become obscured with hyperbolae. To a certain extent, we
can consider SAR (and
seismic migration techniques) a process of focusing these
hyperbolae back to point tar-
gets.
z 0=
yi zi,( )
0 y,( )
z
y
z zi2 yi y–( )
2+=
Figure 3.6: Graphical Representation of How Target Hyperbola
Occurs
Ri
Ri zi2 yi y–( )
2+=
Riz Ri=
z Ri zi2 yi y–( )
2+= =
z 0≥
z2 y yi–( )2– zi
2=
24
-
3.3 The Stripmap SAR Algorithm
The Stripmap SAR derivation (Soumekh [6]) begins by considering
point targets
located in a 2 dimensional space with the ith target having
coordinates and some
measure of the targets reflectivity, . The general setup is
shown in Figure 3.7.
We assume that the radar emits a pulse at all points on the
axis. In the practi-
cal situation we will clearly not have this (the radar will emit
a pulse at a set of evenly
spaced points on the axis), however, for the derivation, we
assume a continuous vari-
able. Assuming a lossless media, we then model the received
signal as:
(3.13)
where is the speed of light in the surrounding medium, and as is
usual in this deriva-
tion, we have neglected the attenuation term associated with
each point. This is justified
by the much greater importance of the phase terms. The initial
modeling equation has bur-
n
yi zi,( )
ρi
y
0 y,( )
y1 z1,( )
y2 z2,( )
yi zi,( )
z
yn zn,( )
Figure 3.7:Stripmap SAR Setup
p t( ) y
y
s y t,( )
s y t,( ) ρip t2 zi
2 yi y–( )2+
cm--------------------------------------–
⎝ ⎠⎜ ⎟⎛ ⎞
i 1=
n
∑=
cm
25
-
ied in it many assumptions about the system. To maintain the
flow of the derivation, these
details are dealt with in section 3.4.
If the Fourier Transform, in the time domain, is taken of
equation (3.13), we get
, (3.14)
where is the wavenumber for lossless media and is the FT of the
trans-
mitted pulse. Here, and for the rest of this derivation, we have
utilized a ‘-’ sign in the for-
ward fourier transform for all variables.
The next step is to take the FT in the spatial domain, . To
solve the resulting inte-
gral, we make use of the method of stationary phase [16], and
arrive at:
. (3.15)
Here, we have neglected another amplitude term , which
results
from the use of the method of stationary phase. It is important
to note that the phase term
in (3.15) is linear in both and .
We now change flow in this derivation and consider how we would
define the
‘ideal’ image for these point targets. A reasonable ‘ideal’
image would be:
, (3.16)
where is the usual two dimensional Dirac delta function. The
spatial FT of
this image in both the and directions is:
. (3.17)
If we compare equations (3.17) and (3.15), we can identify
, (3.18)
where we make use of the so-called spatial frequency mapping
equation:
S y ω,( ) 12π------⎝ ⎠⎛ ⎞
12---
P ω( ) ρiej– 2k zi
2 yi y–( )2+
i 1=
n
∑=
k ω cm⁄= P ω( )
ky
S ky ω,( )1
2π------⎝ ⎠⎛ ⎞P ω( ) ρie
j– 4k2 ky2– zi j– kyyi
i 1=
n
∑=
e j π 4⁄( )– 4k2 ky2–( )⁄
yi zi
go y z,( ) ρiδ y yi z zi–,–( )i 1=
n
∑=
δ y yi z zi–,–( )
y z
Go ky kz,( )1
2π------⎝ ⎠⎛ ⎞ ρie
j– kyyi j– kzzi
i 1=
n
∑=
Go ky kz,( )S ky ω,( )
P ω( )--------------------=
26
-
. (3.19)
By taking the spatial IFT of (3.18), we can construct the
imaging equation as
(3.20)
where is after we have utilized the spatial frequency mapping
equation
(3.19).
In summary, what this derivation shows is that to build the
focused image, we take our
collected data, , take the FT in both the and directions, divide
by the FT of the
pulse , then take the IFT in the and directions. We also require
a scal-
ing term to make this equation a true FT, but this factor is
unimportant for imaging pur-
poses. In the case of SFCW radar, over all frequencies of
interest, and we
collect data in the domain and therefore we can begin our image
construction at
equation (3.14).
3.3.1 The Interpolation Problem in Stripmap SARWhile the above
derivation was done with continous variables, in practical
situa-
tions we have only discrete variables. Specifically, the radar
will collect a set of evenly
spaced discrete points in the domain which leads to a set of
evenly spaced points in the
domain, when utilizing the efficient Fast Fourier Transform
(FFT). We also collect data
at a set of evenly spaced discrete points in the domain (i.e. )
and this leads to a
set of evenly spaced discrete values of .
In order to evaluate the imaging equation, (3.20), we require
that our data lies in the
and domain. To shift the data in the to the domain we utilize
the
spatial frequency mapping equation (3.19). Due to its
non-linearity, (3.19) takes what was
evenly spaced points in the domain and creates a set of unevenly
spaced points in the
domain. Efficient evaluation of (3.20) using the FFT requires
evenly spaced points in
both the and domains. In order to achieve the evenly spaced
points in the
domain, interpolation is required.
kz 4k2 ky
2– 4ω2
cm2
--------- ky2–= =
FSAR y z,( )1
2π------
S ky kz,( )P ω( )
--------------------ej kyy kzz+( ) kyd kzd
∞–
∞
∫∞–
∞
∫=
S ky kz,( ) S ky ω,( )
s y t,( ) ω kyP ω( ) ky kz 1 2π( )⁄
P ω( ) 1=
S y ω,( )
t
ω
y y n∆y=
ky
ky kz ky ω,( ) ky kz,( )
ω
kzkz ky kz
27
-
While at first this problem may seem trivial, improper
consideration of the interpo-
lation problem will result in non or poorly focused images. In
addition, this interpolation
step ends up being the slowest part of the Stripmap SAR
algorithm. In order to avoid con-
fusion in the basic understanding of the Stripmap SAR algorithm,
a detailed exposition of
the interpolation process is given in Appendix A.
3.3.2 Graphical Representation of The Stripmap SAR AlgorithmA
graphical representation of the Stripmap SAR algorithm for at the
2-D problem is
presented in Figure 3.8.
3.4 Electromagnetic Assumptions in Stripmap SAR
In the Stripmap SAR derivation just presented no electromagnetic
considerations
were made. In this section, we will provide a list of
electromagnetic assumptions that are
made in the algorithm which we hope will provide an alternate
way of seeing the Stripmap
SAR algorithm. To the author’s knowledge, this has not been
published previously. We
begin by considering equation (3.13), which provides the initial
model for the Stripmap
SAR algorithm.
The most important electromagnetic assumption made in (3.13) is
that we model our
scatterers as isotropic point scatterers only. This is
equivalent to ignoring all interactions
s y t,( )
s y t,( )2-D Fourier Transform
S ky ω,( )
S ky ω,( )
Interpolation/Mapping
kz 4k2 ky
2–=
2-D Inverse FTS ky kz,( )
P ω( )-------------------- FSAR y z,( )
FSAR y z,( ) Division1 P ω( )( )⁄
Figure 3.8: Graphical Representation of the 2-D Stripmap SAR
Algorithm
28
-
both between, and within scatterers, which might be called a
first-order scattering approx-
imation. The justification given in the literature [6] for the
point target approach to the der-
ivation of the Stripmap SAR algorithm is that larger targets can
be made up of a
summation of these point targets. This assumption may not be
valid, but for practical situ-
ations seems to work reasonably well (i.e. it does provide
focused images).
Secondly, we assume that we have knowledge of the speed of light
in the surround-
ing medium, which is equivalent to knowing the material
parameter of the medium.
Most practical media have . We also assume a lossless medium
(conductivity is
zero).
In addition, we ignore the decay associated with EM fields. In
other words,
only phase information is used in this modelling equation.
We can also notice that equation (3.13) does not make any
reference to the vectorial
character of EM fields. That is, it assumes a scalar wavefield.
We can interpret this as an
assumption that the radar system radiates and receives only one
of the 6 (rectangular) sca-
lar components of the full vector wavefield. This assumption is
identical to the one made
when we apply seismic imaging techniques to the EM problem, and
in practical situations
this approximation does work (i.e. the algorithm provides
focused images) because anten-
nas usually radiate and receive only one vector component of the
EM field.
3.4.1 The Vector Wavenumber in Stripmap SARIn this section, we
take a closer look at the spatial frequency mapping equation,
equation (3.19). From section 2.2, we know that in the two
dimensional problem consid-
ered in the Stripmap SAR derivation, the vector wavenumber
should satisfy the equation:
. (3.21)
However, from (3.19), we see that we have the entirely
unsatisfying result that
. (3.22)
ε
µ µo=
1 R⁄
ky2 kz
2+ k2 ωcm
2------
2= =
ky2 kz
2+ 4ω2
cm2
---------=
29
-
After an analysis of the Stripmap SAR derivation procedure, we
can note that this
discrepancy occurs because we have to take into account both the
forward and return time
of the EM wave. The problem can be solved by applying the
exploding source model.
3.4.2 The Exploding Source ModelIn order to finish the EM
interpretation of Stripmap SAR and to later compare Strip-
map SAR and the seismic based Frequency-Wavenumber migration
(which is covered in
chapter 4), we first must apply a sum of plane waves explanation
of Stripmap SAR. To
accomplish this, we first apply the exploding source model to
Stripmap SAR.
The exploding source model was first introduced by Claerbout
[20]. It is fundamen-
tal to the solution of wave equation based migration techniques
discussed in chapter 4. In
the exploding source model it is assumed that the scattered
field originates from sources
located at the scatters. At time these sources ‘explode’ and
send travelling waves to
the detectors at the surface. To adapt to this new model, we
must replace the velocity of
propagation in the medium, , with half its original value:
. (3.23)
We can then show that equation (3.13) is equivalent to each
point source radiating a plane
wave in all directions, with a varying amplitude term. This is
explored in the next section.
3.4.3 A Plane Wave Interpretation of Stripmap SAR
AlgorithmConsider the phasor form of a general scalar plane wave in
the frequency domain:
, (3.24)
where (i.e. we are using the exploding source model), and is the
unit normal
in the direction of travel of the plane wave,. In this
consideration, we have centered our
coordinate system on the point scatterer, and the antenna
located at the point . The
t 0=
cm
vmcm2
------=
ψ e j– k R⋅ e j– kn R⋅= =
k ω vm⁄= n
yb zb,( )
30
-
problem setup is shown in Figure 3.9. We now assume that the
wave emitted from the
point source (via the exploding source model) to be a plane wave
by the time it reaches the
antenna. We can describe the unit normal of this wave, , as:
. (3.25)
Thus vector wavenumber is given by . The scalar wave is received
at the
point
. (3.26)
We can then write the received signal as
, (3.27)where takes into account the pulse shape and the
reflectivity. If we
note that , and that is simply the distance from the
scatterer to the antenna, we can see that equation (3.13), the
initial modelling equation for
stripmap SAR, can be considered as a sum of scalar plane waves
emanating from each
y
y yb– zb–,( )
0 0,( )
z
yb
n
zb
receiver postion
R
Figure 3.9: Plane Wave Interpretation of Stripmap SAR
n
ny yb–( )ây zbâz–
zb2 y yb–( )
2+----------------------------------------=
k kn=
R y yb–( )ayˆ zbâz–=
S ω y,( )
S ω y,( ) L ω y,( )ej– k
y yb–( )ây zbâz–
zb2 y yb–( )
2+--------------------------------------- y yb–( )ay
ˆ zbâz–•
L ω y,( )ej– k zb
2 y yb–( )2+
= =L ω y,( ) ρP ω( )=
k ω vm⁄ 2ω c⁄= = zb2 y yb–( )
2+
31
-
point target at half the actual medium velocity. To reiterate,
(3.13) makes the approxima-
tion that the receiver is in the far field (i.e. the signal is a
plane wave) and considers only
phase differences in the received signal.
32
-
Chapter 4Seismic Migration
This chapter begins by describing seismic migration techniques.
It begins with the
basic geometric method of Hyperbolic Summation, then introduces
the wave field migra-
tion algorithms of Kirchhoff Migration and Frequency Wavenumber
Migration. The
exploding source model is required for the wave field
techniques, and is described in sec-
tion 4.1.
Migration is the term used by geophysicists to describe the
process of focusing the
basic B-scan images to more closely resemble the physical target
dimensions. These
images arise from a seismic system similar to a GPR, but
radiating sound waves instead of
electromagnetic waves. Early seismic algorithms, such as
Hyperbolic Summation, were
based on a geometric approach and paid little attention to the
physics of seismic wave
propagation. More advanced techniques of migration based on the
scalar wave equation
were introduced in the late 1970’s and early 1980’s. A good
overview of the more
advanced techniques is given in Berkhout [19].
The same three basic assumptions made in the Stripmap SAR
algorithm are made
for seismic algorithms. Once again, migration techniques model
point scatterers only.
Interactions inside, and between targets are ignored. We again
assume a lossless ground
and knowledge of the ground constitutive parameters (i.e. we
assume knowledge of the
velocity of propagation in the medium). The third assumption
that only one EM field com-
ponent is radiated and received is implicit in the application
of seismic algorithms to an
EM problem, as seismic algorithms are based on the scalar wave
equation.
The migration methods in this chapter are presented in 3
dimensions, but all are eas-
ily modified to two dimensional problems. Much of this series of
derivations is based on
Scheers [18].
33
-
4.1 The Exploding Source Model
The wave equation based seismic algorithms of Kirchhoff
Migration and Fre-
quency-Wavenumber migration require the application of the
exploding source model,
discussed previously in section 3.4.2. Essentially, we replace
the actual medium velocity
with half its true value ( ).
Under the exploding source model, wavefield migration consists
of two basic con-
cepts:
1. Backward extrapolation of the received signal to the
exploding sources.2. Defining the image as the backward
extrapolated wave field at time .
In other words, migration consists of back propagating the
received wave front to
the instant the targets ‘explode’. The image is then the scalar
field at the instant before it
begins to propagate.
vm cm 2⁄=
t 0=
34
-
The geometry of the exploding source model is shown in Figure
4.1. Here, we again
see the target hyperbolas seen in SAR imaging. The receiver
locations are physically
located on the plane , and record the data over the entire -
plane. We also assume
a constant velocity throughout the entire medium surrounding the
point target. Viewed
through this diagram, seismic migration is an attempt to take
the data received in the -
plane and extrapolate it back to the - plane.
Here, we may note that we make the same three assumptions used
in Stripmap SAR.
For point targets to ‘explode’ in this fashion, we require no
interactions between point tar-
gets. We again assume knowledge of the velocity in the
surrounding medium, and the
assumption about having only scalar wave fields is inherent to
seismics.
Point Source
t
y
z
t 0=Receiver Locations
Figure 4.1: The Exploding Source Model
z 0= y t
y t
y z
35
-
4.2 Hyperbolic Summation
Hyperbolic Summation (HS), also known as diffraction summation,
is a simple geo-
metric approach to seismic migration. It is presented here
because it provides a good intro-
duction to seismic migration and helps one to understand the
more complicated Kirchhoff
Migration algorithm.
The HS algorithm starts with the assumption that every point in
the desired image
has been created by a diffraction hyperbola, as shown in Figure
4.1. To migrate each point
in the image, , we sum the recorded data (which is the scalar
wavefield at the loca-
tion ) along the calculated hyperbola. The shape of the
hyperbola to sum over will
depend upon the depth of the point to be migrated, and the
medium velocity. If there was a
point scatterer located at , the amplitudes of the hyperbolae
towards a large value.
If there is no point scatterer (i.e. a noise source), the data
will sum towards a much smaller
value.
If we imagine that we have collected (have knowledge of) the
scalar field, on the
plane at a set of discrete points, or where and
. The migrated image can then be expressed as:
(4.1)
where is the distance between the measuring position and the
point to
be migrated , and is the velocity of light in the medium.
As we are concerned with 2-D images in this thesis, we write the
Hyperbolic Sum-
mation algorithm as:
(4.2)
One advantage of this method is that we can select a sub-region
to migrate (by selec-
tively picking our points). However, it does not take the
physics of wave propa-
gation into account, and has been superseded by more advanced
methods. As such, no
images focused by this method are presented.
x y z, ,( )
z 0=
x y z, ,( )
ψ
z 0= ψ xi yj z, , 0 t,=( ) i 1 2…I,=
k 1 2…K,=
FHS x y z, ,( ) ψ xi yk z, , 0 t2Ri k,
cm------------=,=⎝ ⎠
⎛ ⎞
k 1=
K
∑i 1=
I
∑=
Ri j, xi yj z, , 0=( )
x y z, ,( ) cm
FHS y z,( ) ψ yk z, 0 t2Rkcm
---------=,=⎝ ⎠⎛ ⎞
k 1=
I
∑=
x y z, ,( )
36
-
4.3 Kirchhoff Migration
Kirchhoff Migration (KM), which is also known as reverse-time
wave equation
migration or wave field extrapolation, is based on an integral
solution of the scalar wave
equation:
. (4.3)
As per the exploding source model, . The boundary conditions on
the scalar
wave equation specify on the local ground surface (our collected
data), and also specify
that as The diagram associated with this problem is shown in
Figure
4.2. In this diagram, we have the associated quantities:
• is the location of the observer• is the time of the observer•
is the location of the point source over which we integrate
ψ r t,( )∇2 1vm
2--------
t22
∂
∂ ψ r t,( )– 0=
vm cm 2⁄=
ψ
ψ r t,( ) 0→ r ∞→
S'
n'
z 0=Measurement Surface
Boundary at infinity
V'r r'
r r'–θ
Figure 4.2: Kirchhoff Migration Coordinate System
rtr'
37
-
• is the time of the source point• is the surface containing all
the sources, consisting of the data plane and the infinite
hemispheric half-space in .• is the outward normal to • is the
scalar wave field at location and time • is the scalar wave field
that results from a source at point and time .
The Green’s Function is chosen to satisfy the same wave
equation, but with
Dirichlet Conditions on the ground surface ( on ). Using the
image principle, our
Green’s function is:
(4.4)
where is the basic free space Green’s function:
. (4.5)
With this Green’s Function we now write the Kirchhoff Integral
(equation (2.40))
[13] as
. (4.6)
Our choice of G specified that, on ,
(4.7)
and we note that
. (4.8)
From these, we can write the scalar wave field as
, (4.9)
where on the ground surface. This integral is known as the
Rayleigh integral [5].
Taking into account the following identity for the Green’s
function:
, (4.10)
the scalar wave field can be written as
. (4.11)
t'S'
z 0<n' S'ψ r t,( ) r tψ r' t',( ) r' t'
G 0= S′
G r t r' t',,( ) g x y z t x' y' z' t', , ,, , ,( ) g x y z t x'
y' z– ' t', , ,, , ,( )–=
g r t r' t',,( )
g r t r' t',,( ) 14π------δ t t'– r r'– v⁄+( )
r r'–----------------------------------------------=
ψ r t,( ) ψ r′ t,( )n′∂∂ G r t r′ t′,,( ) G r t r′ t′,,( )
n′∂∂ ψ r′ t′,( )–
S′∫° S′d t′d
t′∫–=
S'
G r t r' t',,( ) 0=
n'∂∂ G r t r' t',,( ) 2
n'∂∂ g r t r' t',,( )=
ψ r t,( ) 2– ψ r' t',( )z'∂∂ g r t r' t',,( )
S'∫° S'd t'd
t'∫=
n' z– '=
z'∂∂ g r t r' t',,( )
z∂∂ g r t r' t',,( )–=
ψ r t,( ) 2–z∂∂ ψ r' t',( ) g r t r' t',,( )( )
S'∫° S'd t'd
t'∫=
38
-
This equation gives us a solution for the scalar wavefield for
all times at any loca-
tion. What it tells us is that, in order to find the scalar
wavefield at all locations for all
time, we take our collected data on the plane, , take the
derivative (in the
direction) of the free space Green’s function, then integrate
over both time and the
surface.
Different practical implementations of Kirchhoff Migration will
vary depending on
the approximation used for the derivative of the free space
Green’s function,
.
For the first order approximation, we can write (utilizing the
Taylor expansion):
, (4.12)
then, utilizing the identity
, (4.13)
we can write
(4.14)
and
. (4.15)
Thus,
. (4.16)
In some literature (Yilmaz [21]) the second order term is taken
into
account In this thesis only the first order approximation is
utilized.
The term can be re-written as
, (4.17)
where we have shown the angle in Figure 4.2. Placing the
approximation for into
(4.11) we arrive at:
z 0= ψ r' t',( ) z
z 0=
z∂∂ g r t r' t',,( )
z∂∂g 1
4π------ 1
r r'–--------------
z∂∂ δ t t'– r r'– vm⁄+( ) O
1r r'– 2-----------------⎝ ⎠⎛ ⎞+=
z∂∂ F G z( )( ) F′ G z( )( ) zd
dG=
z∂∂ δ t t'– r r'– vm⁄+( ) δ′ t t'– r r'– vm⁄+( )
14πvm-------------
z∂∂ r r'–=
t′∂∂ δ t t'– r r'– vm⁄+( ) δ′ t t'– r r'– vm⁄+( ) 1–( )=
z∂∂g 1–
4πvm r r'–----------------------------
t′∂∂ δ t t'– r r'– vm⁄+( )
d r r'–dz
------------------ O 1r r'– 2-----------------⎝ ⎠⎛ ⎞+=
O 1 r r'– 2⁄( )
d r r'– dz⁄
d r r'–dz
------------------ z z'–r r'–-------------- θ( )cos= =
θ g
39
-
, (4.18)
then utilizing the sifting property of the Dirac delta
function:
. (4.19)
Finally, the image is the wave field at time , so we may
write
. (4.20)
The term, once taken into the discrete domain, is exactly the
same as the
term in Hyperbolic summation. Thus, KM consists in summing over
theoretical
hyperbolae (in time) in exactly the same manner as HS. However,
there are three differ-
ences observed with these equations. The first is that we take
into account the spreading
losses of spherical waves in the term. Next is the term, and the
third is
that we first take the derivative, in the time domain, of the
received data. This means that
we must calculate the time derivative of each A-scan before we
perform the hyperbolic
summation. In the frequency domain, this involves a simple
multiplication of the data by
.
All experimental data collected in this thesis was collected
along a single line at dis-
crete points , . The imaging equation for KM in this 2-D case
becomes:
. (4.21)
where and are now in two dimensions.
In addition to our knowledge of the scalar wave field being
discrete in , our knowl-
edge of the scalar wave field is discrete in time as well.
Almost invariably the time
will not fall on an exact discrete time-domain data point. Thus,
interpola-
tion of some type is required for the implementation of this
method. In this thesis, we have
used simple nearest-neighbour interpolation to select the
time-domain data point.
ψ r t,( ) 12πvm------------- ψ r' t',( )
t'∂∂ δ t t'– r r'– vm⁄+( )
θ( )cosr r'–
----------------S'∫° S'd t'd
t'∫≈
ψ r t,( ) 12πvm-------------
t'∂∂ ψ r' t r r'– vm⁄+,( )
θ( )cosr r'–
----------------dS'S'∫°≈
t 0=
FKM x y z, ,( )1
2πvm-------------
t'∂∂ ψ x' y' z' 0 r r'– vm⁄,=, ,( )
θ( )cosr r'–
----------------dS'S'∫°=
r r'– vm⁄
2Rj k, cm⁄
1 r r'–⁄ θ( )cos
jω
yk k 1 2 …K, ,=
FKM y z,( )1
2πvm-------------
t'∂∂ ψ y'i z' 0 r r'– vm⁄,=,( )
θ( )cosr r'–
----------------
i 1=
I
∑=
r r′
y
t r r'– vm⁄=
40
-
A graphical representation of the implementation of the
algorithm is given in Figure
4.3.
4.4 Frequency-Wavenumber Migration
Frequency-Wavenumber (F-K), or Stolt Migration, was first
developed by R.H. Stolt
in 1979 [22]. It is also based on the exploding source model and
the scalar wave equation.
The final result closely resembles the final form of the
Stripmap SAR algorithm. A thor-
ough comparison of these two algorithms is completed in section
5.1.2. Also important is
that F-K migration is theoretically identical to KM (see section
5.1.1). The F-K migration
algorithm is presented in a sum of plane waves format.
F-K migration begins by considering a general plane wave of the
form shown in
(2.29). This plane wave obeys the time-domain scalar wave
equation for lossless media:
, (4.22)
and the rectangular components of the vector wavenumber obey the
separation equation,
. (4.23)
t'∂∂ ψ y'i z' 0 r r'– vm⁄,=,( )
ψ y'i z', 0 t,=( )
t'∂∂ Select Subset of Data
t' r r'–vm--------------=
Multiply1
2πvm------------- θ( )cos
r r'–----------------
SummationOver y'i
FKM y z,( )
Figure 4.3: Graphical Representation of 2-D KM
ψ∇2 1vm
2--------
t2
2
∂
∂ ψ– 0=
kx2 ky
2 kz2+ + k2 ω2 vm
2⁄= =
41
-
Accordingly, if we select any three of the components or the
fourth becomes
fixed.
It has been shown (Stratton [9], p.363) that we can represent an
arbitrary wave func-
tion as a sum of plane waves:
, (4.24)
where is an amplitude function of any three of the Fourier
variables.
In GPR, we measure the field on the plane: and
therefore we have
. (4.25)
Noting that this is a FT (with a positive sign in the forward FT
for spatial variables and a
negative sign for time), we can determine the unknown amplitude
function as
. (4.26)
Thus, is simply the FT of the collected data, in the , and
domains.
Utilizing the basic concepts of the exploding source model, the
image is
and,
.(4.27)
Where we have placed a positive sign in the exponentail because
the integrals are from
to . This facilitates later comparisons to Stripmap SAR
imaging.
This equation describes a focused image, however it is not in
the convenient form of
an IFT at this time. In order to eventually utilize the
efficiency of the FFT algorithm, we
make a change of variables from to .
kx ky kz, , ω
ψ x y z t, , ,( )
ψ x y z t, , ,( ) 12π------⎝ ⎠⎛ ⎞
32---
P kx ky ω, ,( )ej– kxx kyy kzz+ +( ) jωt+ kxd kyd ωd
∞–
∞
∫∞–
∞
∫∞–
∞
∫=
P kx ky ω, ,( )
z 0= ψ x y 0 t, , ,( ) p x y t, ,( )=
p x y t, ,( ) 12π------⎝ ⎠⎛ ⎞
32---
P kx ky ω, ,( )ej– kxx kyy+( ) jωt+ kxd kyd ωd
∞–
∞
∫∞–
∞
∫∞–
∞
∫=
P kx ky ω, ,( )1
2π------⎝ ⎠⎛ ⎞
32---
p x y t, ,( )ej kxx kyy+( ) j– ωt xd yd td
∞–
∞
∫∞–
∞
∫∞–
∞
∫=
P kx ky ω, ,( ) p x y t, ,( ) kx ky ω
ψ x y z 0, , ,( )
FFK x y z, ,( ) ψ x y z 0, , ,( )1
2π------⎝ ⎠⎛ ⎞
32---
P kx ky ω, ,( )ej kxx kyy kzz+ +( ) kxd kyd ωd
∞–
∞
∫∞–
∞
∫∞–
∞
∫= =
∞– ∞
ω kz
42
-
Noting from equation (4.23):
, (4.28)
then
, (4.29)
so
. (4.30)
The modified F-K imaging equation then becomes
. (4.31)
Where is mapped from the domain to the domain.
In this form our final image is represented as a FT, which
whe